Sorry - I can't resist....
"I have no idea what am I doing wrong."
Make sure that your life insurance is paid up. Better yet, you really should start by controlling an LED or a low voltage lamp before you deal with a 120 V load. You might also consider using the pricy but much safer 'Powerswitch tail' (http://www.adafruit.com/products/268).
The next best option would be to use a solid state relay in place of most of your components.
Otherwise ...
The relay terminal connections in your upper pair of diagrams do not agree with one another although either will work. In the top half you have the common 'C' terminal connected to the light bulb and in the lower half you have it connected to the AC supply. By the way, this is the lower right terminal in your third diagram, the one you have incorrectly shown with a 'ground' symbol. In the top half you have the normally open 'NO' terminal connected to the AC supply and in the lower half you have it connected to the light bulb. This is the terminal with the red check mark in the third diagram.
Get the lower part of your upper diagram working - that is get the relay to control the light. If you connect 5V between the connections labeled 'Pin 12' and 'GND' the lamp should light up.
Now add the other parts to implement the upper part of the upper diagram. Don't overlook the fact that the battery needs to have it's upper end connected to GND. When you connect the terminal labeled 'I/O pin' to the +5 Volt end of the battery the relay should energize and turn the light on.
Now you can connect the Arduino.
Don
[Edit] I just noticed that this really belongs in the 'General Electronics' section of the forum, perhaps a moderator can move it.